arXiv subjects

Robert Weston

Publications and source records attributed to Robert Weston.

At least 19 recordsLinked to original sources

arXiv2025

Cyclic Representations of $U_q(\hat{\mathfrak{sl}}_2)$ and its Borel Subalgebras at Roots of Unity and Q-operators

We consider the cyclic representations $Ω_{rs}$ of $ U_q(\widehat{\mathfrak{sl}}_2)$ at $q^N=1$ that depend upon two points $r,s$ in the chiral Potts algebraic curve. We show how $Ω_{rs}$ is related to the tensor product $ρ_r\otimes \barρ_s$ of two representations of the upper Borel subalgebra of $U_q(\widehat{\mathfrak{sl}}_2)$. This result is analogous to the factorization property of the Verma module of $U_q(\widehat{\mathfrak{sl}}_2)$ at generic-$q$ in terms of two q-oscillator representation of the Borel subalgebra - a key step in the construction of the Q-operator. We construct short exact sequences of the different representations and use the results to construct Q operators that satisfy TQ relations for $q^N=1$ for both the 6-vertex and $τ_2$ models.

math-ph↗

arXiv2024

A Q-operator for open spin chains II: boundary factorization

One of the features of Baxter's Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang-Baxter equation associated to particular infinite-dimensional representations). To have such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang-Baxter equation) associated to these representations. In the case of quantum affine $\mathfrak{sl}_2$ and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.

math-ph↗

arXiv2022

Fast-MbyM: Leveraging Translational Invariance of the Fourier Transform for Efficient and Accurate Radar Odometry

Masking By Moving (MByM), provides robust and accurate radar odometry measurements through an exhaustive correlative search across discretised pose candidates. However, this dense search creates a significant computational bottleneck which hinders real-time performance when high-end GPUs are not available. Utilising the translational invariance of the Fourier Transform, in our approach, f-MByM, we decouple the search for angle and translation. By maintaining end-to-end differentiability a neural network is used to mask scans and trained by supervising pose prediction directly. Training faster and with less memory, utilising a decoupled search allows f-MByM to achieve significant run-time performance improvements on a CPU (168%) and to run in real-time on embedded devices, in stark contrast to MByM. Throughout, our approach remains accurate and competitive with the best radar odometry variants available in the literature -- achieving an end-point drift of 2.01% in translation and 6.3deg/km on the Oxford Radar RobotCar Dataset.

cs.RO↗

arXiv2020

A Q-operator for open spin chains I: Baxter's TQ relation

We construct a Q-operator for the open XXZ Heisenberg quantum spin chain with diagonal boundary conditions and give a rigorous derivation of Baxter's TQ relation. Key roles in the theory are played by a particular infinite-dimensional solution of the reflection equation and by short exact sequences of intertwiners of the standard Borel subalgebras of $U_q(\widehat{\mathfrak{sl}_2})$. The resulting Bethe equations are the same as those arising from Sklyanin's algebraic Bethe ansatz.

math-ph↗

arXiv2019

Combinatorial solutions to the reflection equation

We use ring-theoretic methods and methods from the theory of skew braces to produce set-theoretic solutions to the reflection equation. We also use set-theoretic solutions to construct solutions to the parameter-dependent reflection equation.

math.RA↗

arXiv2018

Lattice Supersymmetry in the Open XXZ Model: An Algebraic Bethe Ansatz Analysis

We reconsider the open XXZ chain of Yang and Fendley. This model possesses a lattice supersymmetry which changes the length of the chain by one site. We perform an algebraic Bethe ansatz analysis of the model and derive the commutation relations of the lattice SUSY operators with the four elements of the open-chain monodromy matrix. Hence we give the action of the SUSY operator on off-shell and on-shell Bethe states. We show that this action generally takes one on-shell Bethe eigenstate to another. The exception is that a zero-energy vacuum state will be a SUSY singlet. The SUSY pairings of Bethe roots we obtain are analogous to those found previously for closed chains by Fendley and Hagendorf by analysing the Bethe equations.

math-ph↗

arXiv2016

Conserved Currents in the Six-Vertex and Trigonometric Solid-On-Solid Models

We construct quasi-local conserved currents in the six-vertex model with anisotropy parameter $η$ by making use of the quantum-group approach of Bernard and Felder. From these currents, we construct parafermionic operators with spin $1+iη/π$ that obey a discrete-integral condition around lattice plaquettes embedded into the complex plane. These operators are identified with primary fields in a $c=1$ compactified free Boson conformal field theory. We then consider a vertex-face correspondence that takes the six-vertex model to a trigonometric SOS model, and construct SOS operators that are the image of the six-vertex currents under this correspondence. We define corresponding SOS parafermionic operators with spins $s=1$ and $s=1+2iη/π$ that obey discrete integral conditions around SOS plaquettes embedded into the complex plane. We consider in detail the cyclic-SOS case corresponding to the choice $η=iπ(p-p')/p$, with $p'<p$ coprime. We identify our SOS parafermionic operators in terms of the screening operators and primary fields of the associated $c=1-6(p-p')^2/pp'$ conformal field theory.

math-ph↗

arXiv2015

Theta function solutions of the qKZB equation for a face model

We consider the quantum Knizhnik-Zamolodchikov-Bernard equation for a face model with elliptic weights, the SOS model. We provide explicit solutions as theta functions. On the so-called combinatorial line, in which the model is equivalent to the three-colour model, these solutions are shown to be eigenvectors of the transfer matrix with periodic boundary conditions.

math-ph↗

arXiv2015

Discrete Holomorphicity in the Chiral Potts Model

We construct lattice parafermions for the $Z(N)$ chiral Potts model in terms of quasi-local currents of the underlying quantum group. We show that the conservation of the quantum group currents leads to twisted discrete-holomorphicity (DH) conditions for the parafermions. At the critical Fateev-Zamolodchikov point the parafermions are the usual ones, and the DH conditions coincide with those found previously by Rajabpour and Cardy. Away from the critical point, we show that our twisted DH conditions can be understood as deformed lattice current conservation conditions for an underlying perturbed conformal field theory in both the general $N\geq 3$ and $N=2$ Ising cases.

math-ph↗

arXiv2012

Exact and Scaling Form of the Bipartite Fidelity of the Infinite XXZ Chain

We find an exact expression for the bipartite fidelity f=| '|^2, where |vac> is the vacuum eigenstate of an infinite-size antiferromagnetic XXZ chain and |vac>' is the vacuum eigenstate of an infinite-size XXZ chain which is split in two. We consider the quantity -ln(f) which has been put forward as a measure of quantum entanglement, and show that the large correlation length xi behaviour is consistent with a general conjecture -ln(f) ~ c/8 ln(xi), where c is the central charge of the UV conformal field theory (with c=1 for the XXZ chain). This behaviour is a natural extension of the existing conformal field theory prediction of -ln(f) ~ c/8 ln(L) for a length L bipartite system with 0<< L <<xi.

math-ph↗

arXiv2011

Correlation Functions and the Boundary qKZ Equation in a Fractured XXZ Chain

We consider correlation functions of the form ', where |vac> is the vacuum eigenstate of an infinite antiferromagnetic XXZ chain, |vac>' is the vacuum eigenstate of an infinite XXZ chain which is split in two, and O is a local operator. The Hamiltonian of the split chain has no coupling between sites 1 and 0 and has a staggered magnetic field at these two sites; it arises from a tensor product of left and right boundary transfer matrices. We find a simple, exact expression for ' and an exact integral expression for general ' using the vertex operator approach. We compute the integral when O=σ^z_1 and find a conjectural expression that is analogous to the known formula for the XXZ spontaneous magnetisation and reduces to it when the magnetic field is zero. We show that correlation functions obey a boundary qKZ equation of a different level to the infinite XXZ chain with one boundary.

math-ph↗

arXiv2010

An Algebraic Setting for Defects in the XXZ and Sine-Gordon Models

We construct defects in the XXZ and sine-Gordon models by making use of the representation theory of quantum affine sl_2. The representations involved are generalisations of the infinite-dimensional, q-oscillator representations used in the construction of Q-operators. We present new results for intertwiners of these representations, and use them to consider both quantum spin-chain Hamiltonians with defects and quantum defects in the sine-Gordon model. We connect specialisations our results with the work of Corrigan and Zambon on type I and type II defects, and present sine-Gordon soliton/defect and candidate defect/defect scattering matrices.

math-ph↗

arXiv2006

The Entanglement Entropy of Solvable Lattice Models

We consider the spin k/2 analogue of the XXZ quantum spin chain. We compute the entanglement entropy S associated with splitting the infinite chain into two semi-infinite pieces. In the scaling limit, we find S ~ c_k/6 (ln(xi))+ln(g)+... . Here xi is the correlation length and c_k=3k/(k+2) is the central charge associated with the sl_2 WZW model at level k. ln(g) is the boundary entropy of the WZW model. Our result extends previous observations and suggests that this is a simple and perhaps rather general way both of extracting the central charge of the ultraviolet CFT associated with the scaling limit of a solvable lattice model, and of matching lattice and CFT boundary conditions.

math-ph↗

arXiv2005

The Vertex-Face Correspondence and Correlation Functions of the Fusion Eight-Vertex Model I: The General Formalism

Based on the vertex-face correspondence, we give an algebraic analysis formulation of correlation functions of the $k\times k$ fusion eight-vertex model in terms of the corresponding fusion SOS model. Here $k\in Z_{>0}$. A general formula for correlation functions is derived as a trace over the space of states of lattice operators such as the corner transfer matrices, the half transfer matrices (vertex operators) and the tail operator. We give a realization of these lattice operators as well as the space of states as objects in the level $k$ representation theory of the elliptic algebra $U_{q,p}(\hat{sl}_2)$.

math.QA↗

arXiv2002

A Generalized Q-operator for U_q(\hat(sl_2)) Vertex Models

In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix of $U_q(\hat{sl}_2)$ over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q-oscillator representations used previously. We derive generalized T-Q relations in which 3 of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q-operator of the 6-vertex model and show the connection with Baxter's expression for the central block of his corresponding operator.

math-ph↗

arXiv2000

Impurity Operators in RSOS Models

We give a construction of impurity operators in the `algebraic analysis' picture of RSOS models. Physically, these operators are half-infinite insertions of certain fusion-RSOS Boltzmann weights. They are the face analogue of insertions of higher spin lines in vertex models. Mathematically, they are given in terms of intertwiners of $U(\hat{sl}_2)_q$ modules. We present a detailed perturbation theory check of the conjectural correspondence between the physical and mathematical constructions in a particular simple example.

math-ph↗

arXiv1999

Vertex Models with Alternating Spins

The diagonalisation of the transfer matrices of solvable vertex models with alternating spins is given. The crystal structure of (semi-)infinite tensor products of finite-dimensional $U_q(\hat{sl}_2)$ crystals with alternating dimensions is determined. Upon this basis the vertex models are formulated and then solved by means of $U_q(\hat{sl}_2)$ intertwiners.

math.QA↗

arXiv1998

Mixing of Ground States in Vertex Models

We consider the analogue of the 6-vertex model constructed from alternating spin n/2 and spin m/2 lines, where $1\leq n<m$. We identify the transfer matrix and the space on which it acts in terms of the representation theory of $U_q(sl_2)$. We diagonalise the transfer matrix and compute the S-matrix. We give a trace formula for local correlation functions. When n=1, the 1-point function of a spin m/2 local variable for the alternating lattice with a particular ground state is given as a linear combination of the 1-point functions of the pure spin m/2 model with different ground states. The mixing ratios are calculated exactly and are expressed in terms of irreducible characters of $U_q(sl_2)$ and the deformed Virasoro algebra.

hep-th↗